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TECHNISCHE UNIVERSIT¨ AT WIEN Entropy method for hypocoercive & non-symmetric Fokker-Planck equations with linear drift

Anton ARNOLD

with Jan Erb

Capri, September 2015

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 1 / 33

degenerate Fokker-Planck equations with linear drift

evolution of probability density f (x, t), x ∈ Rn, t > 0: ft = div

• D∇f + Cx f
• (1)

f (x, 0) = f0(x) D ∈ Rn×n ... symmetric, const in x, degenerate C ∈ Rn×n ... const in x

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 2 / 33

degenerate Fokker-Planck equations with linear drift

evolution of probability density f (x, t), x ∈ Rn, t > 0: ft = div

• D∇f + Cx f
• (1)

f (x, 0) = f0(x) D ∈ Rn×n ... symmetric, const in x, degenerate C ∈ Rn×n ... const in x goals: existence & uniqueness of steady state f∞(x); convergence f (t) t→∞ − → f∞ with sharp rates; complete theory for the equation class (1)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 2 / 33

hypocoercive example – from plasma physics

kinetic Fokker-Planck equation for f (x, v, t) , x, v ∈ Rn: ft + v · ∇xf

free transport

− ∇xV · ∇vf

• influence of potential V (x)

= σ∆vf

diffusion, σ>0

+ ν divv(vf )

• friction, ν>0

steady state: f∞(x, v) = c e

− ν

σ

• |v|2

2 +V (x)

• V (x)... given confinement potential

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 3 / 33

hypocoercive example – from plasma physics

kinetic Fokker-Planck equation for f (x, v, t) , x, v ∈ Rn: ft + v · ∇xf

free transport

− ∇xV · ∇vf

• influence of potential V (x)

= σ∆vf

diffusion, σ>0

+ ν divv(vf )

• friction, ν>0

steady state: f∞(x, v) = c e

− ν

σ

• |v|2

2 +V (x)

• V (x)... given confinement potential

rewritten (with x, v variables): ft = divx,v σ I

• =:D...diffusion

∇x,vf +

• −v

∇xV + ν v

• drift

f

• Anton ARNOLD (TU Vienna)

hypocoercive Fokker-Planck/entropy meth. 3 / 33

Outline:

1 hypocoercivity, prototypic examples 2 review of standard entropy method for non-degenerate

Fokker-Planck equations

3 decay of modified “entropy dissipation” functional 4 mechanism of new method Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 4 / 33

(hypo)coercivity 1

example 1: standard Fokker-Planck equation on Rn: ft = div

• ∇f + x f
• =: Lf . . . symmetric on H := L2(f −1

∞ )

f∞(x) = ce − |x|2

2 ,

ker L = span(f∞)

• L is dissipative, i.e. Lf , f H ≤ 0

∀f ∈ D(L)

• −L is coercive (has a spectral gap), in the sense:

−Lf , f H ≥ f 2

L2(f −1

∞ )

∀ f ∈ {f∞}⊥

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 5 / 33

(hypo)coercivity 2

example 2: ft = div

• D∇f + Cx f
• =: Lf

(2) with degenerate D is degenerate parabolic; (symmetric part of) −L is not coercive.

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 6 / 33

(hypo)coercivity 2

example 2: ft = div

• D∇f + Cx f
• =: Lf

(2) with degenerate D is degenerate parabolic; (symmetric part of) −L is not coercive.

Definition 1 (Villani 2009)

Consider L on Hilbert space H with K = ker L; let ˜ H ֒ → K⊥ (densely) (e.g. H ... weighted L2, ˜ H ... weighted H1). −L is called hypocoercive on ˜ H if ∃ λ > 0, c ≥ 1: e Ltf ˜

H ≤ c e −λtf ˜ H

∀ f ∈ ˜ H

• typically c > 1

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 6 / 33

Steady state of (non)degenerate FP equations:

standard Fokker-Planck equation ft = div(∇f + x f ) : unique steady state f∞(x) = c e −|x|2/2 as a balance of drift & diffusion; sharp decay rate = 1 n = 2: x1 x2 drift diffusion

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 7 / 33

degenerate prototype:

degenerate diffusion (1D Fokker-Planck) + rotation ft = div 1

• =D

∇f + x1−ω x2 ω x1

• =Cx

f

• f∞(x) = c e −|x|2/2

∀ ω ∈ R (unique for ω = 0); sharp decay rate = 1

2 (= min ℜ λC) for fast enough rotation (|ω| > 1 2)

x1 x2 equilibration by drift/diffusion

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 8 / 33

degenerate prototype:

degenerate diffusion (1D Fokker-Planck) + rotation ft = div 1

• =D

∇f + x1−ω x2 ω x1

• =Cx

f

• f∞(x) = c e −|x|2/2

∀ ω ∈ R (unique for ω = 0); sharp decay rate = 1

2 (= min ℜ λC) for fast enough rotation (|ω| > 1 2)

x1 x2 equilibration by drift/diffusion |ω| = 1

2: C has a Jordan block ⇒ (sharp) decay rate = 1 2 − ε

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 8 / 33

degenerate prototype (2):

ft = div 1

• ∇f +

1 −1 1

• x f
• x2-axis: drift characteristics of ˙

x = −Cx tangent to level curve of |x| :

x ’ = − x + y y ’ = − x −4 −3 −2 −1 1 2 3 4 −3 −2 −1 1 2 3 x y Drift characteristic Level curve of P−norm

level curve of “distorted” vector norm √ x · P · x; P =

• 2

−1 −1 2

• Ref: [Dolbeault-Mouhot-Schmeiser] 2015

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 9 / 33

coefficients C, D in Fokker-Planck equation

ft = div

• D ∇f + C x f
• =: Lf

Condition A: No (nontrivial) subspace of ker D is invariant under C⊤. (equivalent: L is hypoelliptic.)

Proposition 1

Let Condition A hold. a) Let f0 ∈ L1(Rd) ⇒ f ∈ C ∞(Rn × R+). [H¨

• rmander 1969]

b) Let f0 ∈ L1

+(Rd)

⇒ f (x, t) > 0, ∀t > 0. (Green’s fct > 0)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 10 / 33

coefficients C, D in Fokker-Planck equation

ft = div

• D ∇f + C x f
• =: Lf

Condition A: No (nontrivial) subspace of ker D is invariant under C⊤. (equivalent: L is hypoelliptic.)

Proposition 1

Let Condition A hold. a) Let f0 ∈ L1(Rd) ⇒ f ∈ C ∞(Rn × R+). [H¨

• rmander 1969]

b) Let f0 ∈ L1

+(Rd)

⇒ f (x, t) > 0, ∀t > 0. (Green’s fct > 0) Condition B: Condition A + let C be positively stable (i.e. ℜ λC > 0) → ∃ confinement potential; drift towards x = 0.

• hypoelliptic + confinement = hypocoercive (for FP eq.)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 10 / 33

steady state

ft = div

• D∇f + Cx f
• (3)

Theorem 2

(3) has a unique (normalized) steady state f∞ ∈ L1(Rn) iff Condition B holds. Then: f∞(x) = cKe − x⊤K−1x

2

. . . non-isotropic Gaussian 0 < K ∈ Rn×n . . . unique solution of 2D = CK + KC⊤ (continuous Lyapunov equation)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 11 / 33

normalization of Fokker-Planck equations

ft = div

• D∇f + Cx f
• with f∞(x) = cKe − x⊤K−1x

2

transformations:

1 y :=

√ K

−1x ⇒

gt = divy ˜ D∇yg + ˜ Cy g

• with g∞(x) = ce − |y|2

2 ,

˜ D = ˜ CS

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 12 / 33

normalization of Fokker-Planck equations

ft = div

• D∇f + Cx f
• with f∞(x) = cKe − x⊤K−1x

2

transformations:

1 y :=

√ K

−1x ⇒

gt = divy ˜ D∇yg + ˜ Cy g

• with g∞(x) = ce − |y|2

2 ,

˜ D = ˜ CS

2 rotation of y ⇒

ˇ D = diag(d1, ..., dk, 0, ..., 0

n−k

) [normalization from now on assumed]

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 12 / 33

review of entropy method: linear symmetric Fokker-Planck equations

evolution of probability density f (x, t), x ∈ Rn, t > 0: ft = div

• D(x) · [∇f + f ∇A(x)]
• =: Lf

f (x, 0) = f0(x); f0 ∈ L1

+(Rn),

• Rn f0 dx = 1

⇒ f (x, t) ≥ 0

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 13 / 33

review of entropy method: linear symmetric Fokker-Planck equations

evolution of probability density f (x, t), x ∈ Rn, t > 0: ft = div

• D(x) · [∇f + f ∇A(x)]
• =: Lf

f (x, 0) = f0(x); f0 ∈ L1

+(Rn),

• Rn f0 dx = 1

⇒ f (x, t) ≥ 0 f∞(x) = e −A(x) . . . (unique) normalized steady state Lf = div

• f∞D(x)∇ f

f∞

• ... symmetric in L2(Rn, f −1

∞ )

D(x) > 0 ... positive definite matrix ∀ x ∈ Rn A(x) ... scalar confinement potential, i.e. A(x) → ∞ as |x| → ∞; idea : A(x) c|x|2

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 13 / 33

admissible relative entropies (for entropy method)

for probability densities f1,2: eψ(f1|f2) :=

• Rn ψ

f1 f2

• f2 dx ≥ 0

... relative entropy ψ : R+

0 → R+

... entropy generators ψ ≥ 0, ψ(1) = 0, ψ′′ > 0, (ψ′′′)2 ≤ 1

2ψ′′ψIV

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 14 / 33

admissible relative entropies (for entropy method)

for probability densities f1,2: eψ(f1|f2) :=

• Rn ψ

f1 f2

• f2 dx ≥ 0

... relative entropy ψ : R+

0 → R+

... entropy generators ψ ≥ 0, ψ(1) = 0, ψ′′ > 0, (ψ′′′)2 ≤ 1

2ψ′′ψIV

examples: 1) ψ1(σ) = σ ln σ − σ + 1 2) ψp(σ) = σp − 1 − p (σ − 1), 1 < p ≤ 2

1 σ ψ(σ) ϕ . . . quadr.

eψ(f1|f2) = 0 ⇐ ⇒ f1 = f2

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 14 / 33

entropy dissipation

Lemma 1

Let f (t) solve Fokker-Planck equation ft = div

• D(x) · [∇f + f ∇A(x)]
• ⇒ d

dt eψ(f (t)|f∞) = −

• Rn ψ′′

f (t) f∞

• ∇⊤f (t)

f∞ · D(x) · ∇f (t) f∞ f∞ dx =: −Iψ(f (t)|f∞) ≤ 0 ... (negative) Fisher information

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 15 / 33

Step1: exponent. decay of entropy dissipation for D≡const

D = const. in x, f∞(x) = e −A(x)

Theorem 3

Let Iψ(f0|f∞) < ∞. Let D, A satisfy a Bakry - Emery condition ∂2A(x) ∂x2 ≥ λ1

>0

D−1 ∀x ∈ Rn (4) ⇒ Iψ(f (t)|f∞) ≤ e −2λ1t Iψ(f0|f∞) , t ≥ 0 A . . . uniformly convex if D = I Ref’s: [Bakry-Emery] 1984/85; [Arnold-Markowich-Toscani-Unterreiter] Comm. PDE 2001

• robust w.r.t. many nonlinear perturbations

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 16 / 33

Step 2: exponential decay of relative entropy for D≡const

Theorem 4

Let D, A satisfy BEC

∂2A(x) ∂x2

≥ λ1D−1 ⇒ eψ(f (t)|f∞) ≤ e −2λ1t eψ(f0|f∞), t ≥ 0

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 17 / 33

Step 2: exponential decay of relative entropy for D≡const

Theorem 4

Let D, A satisfy BEC

∂2A(x) ∂x2

≥ λ1D−1 ⇒ eψ(f (t)|f∞) ≤ e −2λ1t eψ(f0|f∞), t ≥ 0 Proof: from proof of Theorem 3 : d dt I(t) ≤ −2λ1 I(t)

• =−e′(t)
• ∞

t

. . . dt Since I(t), e(t)

t→∞

− − − → 0: d dt e(t) ≤ −2λ1e(t) (5) (+ density argument)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 17 / 33

problem of entropy decay (cp. standard entropy method)

decay of quadratic entropy e2(t) = f (t) − f∞2

L2(f −1

∞ ) :

standard Fokker-Planck equation: non-degenerate → e(t) is convex; entropy dissip. e′(t) < 0 ∀ f = f∞; e′ ≤ −μe possible (with μ > 0)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 18 / 33

problem of entropy decay (cp. standard entropy method)

decay of quadratic entropy e2(t) = f (t) − f∞2

L2(f −1

∞ ) :

standard Fokker-Planck equation: non-degenerate → e(t) is convex; entropy dissip. e′(t) < 0 ∀ f = f∞; e′ ≤ −μe possible (with μ > 0) degenerate prototype ex.: → e(t) is not convex; e′(t) = 0 for some f = f∞; e′ ≤ −μe wrong (in general)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 18 / 33

entropy decay in inhomogeneous Boltzmann equation

simulation of 1+2 D Boltzmann equation: wavy entropy decay − − − Hg(t): relative entropy w.r.t. the global Maxwellian Ref: [Filbet-Mouhot-Pareschi] 2006

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 19 / 33

modified entropy method for degenerate FP equation

e′(t) = 0 for some f = f∞ ⇒ entropy dissipation: d dt eψ = −

• Rn ψ′′

f f∞

• ∇⊤ f

f∞ · D

• ≥0

·∇ f f∞ f∞ dx =: −Iψ(f ) ≤ 0 is “useless” as Lyapunov functional.

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 20 / 33

modified entropy method for degenerate FP equation

e′(t) = 0 for some f = f∞ ⇒ entropy dissipation: d dt eψ = −

• Rn ψ′′

f f∞

• ∇⊤ f

f∞ · D

• ≥0

·∇ f f∞ f∞ dx =: −Iψ(f ) ≤ 0 is “useless” as Lyapunov functional. ⇒ define modified “entropy dissipation” as auxiliary functional: Sψ(f ) :=

• Rn ψ′′

f f∞

• ∇⊤ f

f∞ · P

• >0

·∇ f f∞ f∞ dx ≥ 0 goal: estimate between S(f (t)),

d dt S(f (t)) for “good” choice of P > 0.

Then: P ≥ cP D ⇒ Sψ(f ) ≥ cP Iψ(f ) ց 0

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 20 / 33

modified “entropy dissipation” Sψ(f ) : choice of P

Lemma 2

Let Q be positively stable, i.e. μ := min{ℜ λQ} > 0 .

1 If all λmin

Q

∈ {λ ∈ σ(Q) | ℜλ = μ} are non-defective (i.e. geometric = algebraic multiplicity) ⇒ ∃ P ∈ Rn×n, P > 0 : PQ + Q⊤P ≥ 2μP .

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 21 / 33

modified “entropy dissipation” Sψ(f ) : choice of P

Lemma 2

Let Q be positively stable, i.e. μ := min{ℜ λQ} > 0 .

1 If all λmin

Q

∈ {λ ∈ σ(Q) | ℜλ = μ} are non-defective (i.e. geometric = algebraic multiplicity) ⇒ ∃ P ∈ Rn×n, P > 0 : PQ + Q⊤P ≥ 2μP .

2 If (at least) one λmin

Q

is defective ⇒ ∀ ε > 0 ∃ P = P(ε) > 0 : PQ + Q⊤P ≥ 2(μ − ε)P .

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 21 / 33

modified “entropy dissipation” Sψ(f ) : choice of P

Lemma 2

Let Q be positively stable, i.e. μ := min{ℜ λQ} > 0 .

1 If all λmin

Q

∈ {λ ∈ σ(Q) | ℜλ = μ} are non-defective (i.e. geometric = algebraic multiplicity) ⇒ ∃ P ∈ Rn×n, P > 0 : PQ + Q⊤P ≥ 2μP .

2 If (at least) one λmin

Q

is defective ⇒ ∀ ε > 0 ∃ P = P(ε) > 0 : PQ + Q⊤P ≥ 2(μ − ε)P . Proof: P can be constructed explicitly; e.g. for Q non-defective / diagonalizable: P :=

n

• j=1

zj ⊗ ¯ z⊤

j

; zj ... eigenvectors of Q⊤

• P not unique; but the decay rates μ (or μ − ε) are independent of P.
• application with Q := C .

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 21 / 33

Step 1: exponential decay of auxiliary functional Sψ(f )

Sψ(f ) :=

• Rn ψ′′

f f∞

• ∇⊤ f

f∞ · P · ∇ f f∞ f∞ dx ≥ 0

Proposition 2

μ := min{ℜ λC}. Let f0 satisfy:

• ψ′′

f0 f∞

• ∇ f0

f∞

• 2

f∞ dx < ∞ (∼ weighted H1–seminorm)

1 If all λmin

C

are non-defective ⇒ S(f (t)) ≤ e −2µtS(f0) , t ≥ 0;

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 22 / 33

Step 1: exponential decay of auxiliary functional Sψ(f )

Sψ(f ) :=

• Rn ψ′′

f f∞

• ∇⊤ f

f∞ · P · ∇ f f∞ f∞ dx ≥ 0

Proposition 2

μ := min{ℜ λC}. Let f0 satisfy:

• ψ′′

f0 f∞

• ∇ f0

f∞

• 2

f∞ dx < ∞ (∼ weighted H1–seminorm)

1 If all λmin

C

are non-defective ⇒ S(f (t)) ≤ e −2µtS(f0) , t ≥ 0;

2 If one λmin

C

is defective ⇒ S(f (t), ε) ≤ e −2(µ−ε)tS(f0, ε) , t ≥ 0.

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 22 / 33

Proof of Proposition 2 – modified entropy method

d dt S(f (t)) = −

• ψ′′ f

f∞

• u⊤
• PC + C⊤P
• ≥2µP ... replaces BEC

uf∞ dx −2

• Tr (XY)

≥0

f∞ dx ≤ −2μ S(f (t)) ; u = ∇ f f∞

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 23 / 33

Proof of Proposition 2 – modified entropy method

d dt S(f (t)) = −

• ψ′′ f

f∞

• u⊤
• PC + C⊤P
• ≥2µP ... replaces BEC

uf∞ dx −2

• Tr (XY)

≥0

f∞ dx ≤ −2μ S(f (t)) ; u = ∇ f f∞ use X := ψ′′ ψ′′′ ψ′′′

1 2ψIV

f f∞

• ≥ 0

, since det X = 1 2ψ′′ψIV − (ψ′′′)2 ≥ 0 (for admissible rel. entropies) ;

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 23 / 33

Proof of Proposition 2 – modified entropy method

d dt S(f (t)) = −

• ψ′′ f

f∞

• u⊤
• PC + C⊤P
• ≥2µP ... replaces BEC

uf∞ dx −2

• Tr (XY)

≥0

f∞ dx ≤ −2μ S(f (t)) ; u = ∇ f f∞ use X := ψ′′ ψ′′′ ψ′′′

1 2ψIV

f f∞

• ≥ 0

, since det X = 1 2ψ′′ψIV − (ψ′′′)2 ≥ 0 (for admissible rel. entropies) ; Y :=

• Tr (D∂u

∂x P ∂u ∂x )

u⊤D∂u

∂x Pu

u⊤D∂u

∂x Pu

(u⊤Pu) (u⊤Du)

• ≥ 0 ,

with Cauchy-Schwarz

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 23 / 33

Step 2: exponential decay of relative entropy

Theorem 5

Let f0 satisfy:

• ψ′′

f0 f∞

• |u0|2 f∞ dx < ∞ .

⇒ e(f (t)|f∞) ≤c S(f (t)) ≤ c e −2µtS(f0) , t ≥ 0 ( reduced rate for a defective λmin

C : 2(μ − ε) )

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 24 / 33

Step 2: exponential decay of relative entropy

Theorem 5

Let f0 satisfy:

• ψ′′

f0 f∞

• |u0|2 f∞ dx < ∞ .

⇒ e(f (t)|f∞) ≤c S(f (t)) ≤ c e −2µtS(f0) , t ≥ 0 ( reduced rate for a defective λmin

C : 2(μ − ε) )

Proof: Consider non-degenerate (auxiliary) symmetric FP equation: gt = div

• P
• >0

(∇ g f∞ )f∞

• ;

g∞ = f∞ = c e −|x|2/2 = c e −A(x) (6) It satisfies the Bakry-Emery condition ∂2A

∂x2 = I ≥ λPP−1 .

⇒ convex Sobolev inequality: eψ(g|f∞) ≤ 1 2λP Sψ(g) ∀ g Remark: Sψ(g) is the true entropy dissipation for (6) !

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 24 / 33

(parabolic) regularization of semigroup e Lt

Proposition 3

The H¨

• rmander order

m ∈ [1, n − k] (k =rank D) is the minimum such that

m

• j=0

CjD(C⊤)j ≥ κ I for some κ > 0 . (Existence of m is equivalent to Condition A, i.e. hypoellipticity of L.) ⇒ Sψ(f (t)) ≤ c t−(2m+1) eψ(f0|f∞) , 0 < t ≤ 1 .

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 25 / 33

(parabolic) regularization of semigroup e Lt

Proposition 3

The H¨

• rmander order

m ∈ [1, n − k] (k =rank D) is the minimum such that

m

• j=0

CjD(C⊤)j ≥ κ I for some κ > 0 . (Existence of m is equivalent to Condition A, i.e. hypoellipticity of L.) ⇒ Sψ(f (t)) ≤ c t−(2m+1) eψ(f0|f∞) , 0 < t ≤ 1 . Ref’s:

• Prop. 3 is generalization to all admissible relative entropies of:

[H´ erau] JFA 2007; [Villani] book 2009 (only for quadratic & logarithmic entropies)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 25 / 33

• exp. decay of rel. entropy for ft = div(D∇f + Cx f ) =: Lf

combination of regularization for initial time with Th.5 (entropy decay) ⇒

Theorem 6 (Arnold-Erb 2014)

Let L satisfy Condition B; μ := min{ℜ λC}. ⇒ ∃ c > 0: eψ(f (t)|f∞) ≤ c e −2µteψ(f0|f∞) , t ≥ 0 ( reduced rate for a defective λmin

C : 2(μ − ε) )

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 26 / 33

• exp. decay of rel. entropy for ft = div(D∇f + Cx f ) =: Lf

combination of regularization for initial time with Th.5 (entropy decay) ⇒

Theorem 6 (Arnold-Erb 2014)

Let L satisfy Condition B; μ := min{ℜ λC}. ⇒ ∃ c > 0: eψ(f (t)|f∞) ≤ c e −2µteψ(f0|f∞) , t ≥ 0 ( reduced rate for a defective λmin

C : 2(μ − ε) )

Proof: e(t)

CSI

≤ 1 2λP S(f (t))

decay

≤ 1 2λP e −2µ(t−δ) S(f (δ))

regularization

≤ c(δ) e−2µte (0)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 26 / 33

• exp. decay of rel. entropy for ft = div(D∇f + Cx f ) =: Lf

combination of regularization for initial time with Th.5 (entropy decay) ⇒

Theorem 6 (Arnold-Erb 2014)

Let L satisfy Condition B; μ := min{ℜ λC}. ⇒ ∃ c > 0: eψ(f (t)|f∞) ≤ c e −2µteψ(f0|f∞) , t ≥ 0 ( reduced rate for a defective λmin

C : 2(μ − ε) )

Proof: e(t)

CSI

≤ 1 2λP S(f (t))

decay

≤ 1 2λP e −2µ(t−δ) S(f (δ))

regularization

≤ c(δ) e−2µte (0) Remark: Rate μ is sharp, but constant c is not.

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 26 / 33

kinetic Fokker-Planck eq. with non-quadratic potential

ft + v · ∇xf − ∇xV (x) · ∇vf = σ∆vf + ν divv(vf ) ; x, v ∈ Rn steady state factors in x, v: f∞(x, v) = c e

− ν

σ

• |v|2

2 +V (x)

• Anton ARNOLD (TU Vienna)

hypocoercive Fokker-Planck/entropy meth. 27 / 33

kinetic Fokker-Planck eq. with non-quadratic potential

ft + v · ∇xf − ∇xV (x) · ∇vf = σ∆vf + ν divv(vf ) ; x, v ∈ Rn steady state factors in x, v: f∞(x, v) = c e

− ν

σ

• |v|2

2 +V (x)

• ft = divx,v
• [D + R] ∇x,v

f f∞

• f∞
• ,

D = σ I

• ,

R = σ ν −I I

• independent of x!

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 27 / 33

kinetic Fokker-Planck eq. with non-quadratic potential

ft + v · ∇xf − ∇xV (x) · ∇vf = σ∆vf + ν divv(vf ) ; x, v ∈ Rn steady state factors in x, v: f∞(x, v) = c e

− ν

σ

• |v|2

2 +V (x)

• ft = divx,v
• [D + R] ∇x,v

f f∞

• f∞
• ,

D = σ I

• ,

R = σ ν −I I

• independent of x!

Theorem 7 (Arnold-Erb, Achleitner-Arnold 2015)

Let n = 1; V (x) = ω2

2 |x|2 + ˜

V (x) ... given confinement potential with

• max V ′′(x) −
• min V ′′(x) ≤ ν

⇒ eψ(f (t)|f∞) ≤ c Sψ(f0)e−2κt, t ≥ 0

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 27 / 33

kinetic Fokker-Planck eq. with non-quadratic potential

Proof: Sψ(f ) :=

• R2n ψ′′

f f∞

• ∇⊤ f

f∞ · P

• >0

·∇ f f∞ f∞ dx dv ; which P ? d dt Sψ(f (t))≤−

• ψ′′ f

f∞

• ∇⊤ f

f∞

• Q(x)P + PQT(x)
• ≥2κP ??
• ∇ f

f∞ f∞ dx dv with Q(x) =

• 1

−V ′′(x) ν

• ,

V (x) = ω2

2 |x|2 + ˜

V (x)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 28 / 33

kinetic Fokker-Planck eq. with non-quadratic potential

Proof: Sψ(f ) :=

• R2n ψ′′

f f∞

• ∇⊤ f

f∞ · P

• >0

·∇ f f∞ f∞ dx dv ; which P ? d dt Sψ(f (t))≤−

• ψ′′ f

f∞

• ∇⊤ f

f∞

• Q(x)P + PQT(x)
• ≥2κP ??
• ∇ f

f∞ f∞ dx dv with Q(x) =

• 1

−V ′′(x) ν

• ,

V (x) = ω2

2 |x|2 + ˜

V (x) decay condition: choose P for Q0(x) =

• 1

−ω2 ν

• and κ0 > 0.

⇒ ˜ V ′′ is a small perturbation.

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 28 / 33

local vs. global decay rate in non-symmetric FP equations

decay of logarithmic entropy for the non-symmetric FP equation ft = div 1/4 1

• ∇f +

1/4 x1 − 4 x2 4 x1 + x2

• f
• with f∞(x) = c e −|x|2/2 = e −A(x) :

find e(f (t)|f∞) ≤ c e(0) e −λt

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 29 / 33

local vs. global decay rate in non-symmetric FP equations

decay of logarithmic entropy for the non-symmetric FP equation ft = div 1/4 1

• ∇f +

1/4 x1 − 4 x2 4 x1 + x2

• f
• with f∞(x) = c e −|x|2/2 = e −A(x) :

find e(f (t)|f∞) ≤ c e(0) e −λt standard entropy method: BEC

∂2A ∂x2 = I ≥ λD−1 yields sharp

local decay rate: λl = 1

4;

c = 1 “Hypocoercive method” yields sharp global decay rate: λg = 5

8;

c > 1

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 29 / 33

local vs. global decay rate in non-symmetric FP equations

decay of logarithmic entropy for the non-symmetric FP equation ft = div 1/4 1

• ∇f +

1/4 x1 − 4 x2 4 x1 + x2

• f
• with f∞(x) = c e −|x|2/2 = e −A(x) :

find e(f (t)|f∞) ≤ c e(0) e −λt standard entropy method: BEC

∂2A ∂x2 = I ≥ λD−1 yields sharp

local decay rate: λl = 1

4;

c = 1 “Hypocoercive method” yields sharp global decay rate: λg = 5

8;

c > 1 Lemma: 2D, ∀ admissible P: constant S(0)

2λP is sharp.

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 29 / 33

Why does the “hypocoercive method” work?

algebraic essence: comparison of the spectral gaps of a non-symmetric matrix Q and its symmetric part Qs := 1

2(Q + QT) .

motivation: entropy method operates mostly with quadratic functionals (e.g. e2(f |f∞); Iψ =

• ψ′′( f

f∞ )∇T f f∞ · D · ∇ f f∞ f∞ dx )

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 30 / 33

Why does the “hypocoercive method” work?

algebraic essence: comparison of the spectral gaps of a non-symmetric matrix Q and its symmetric part Qs := 1

2(Q + QT) .

motivation: entropy method operates mostly with quadratic functionals (e.g. e2(f |f∞); Iψ =

• ψ′′( f

f∞ )∇T f f∞ · D · ∇ f f∞ f∞ dx )

Lemma 3

Let Q be positively stable, i.e. μ := min{ℜ λQ} > 0 . ⇒ λmin(Qs) ≤ μ (“typically” strict inequality)

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 30 / 33

Why does the “hypocoercive method” work?

algebraic essence: comparison of the spectral gaps of a non-symmetric matrix Q and its symmetric part Qs := 1

2(Q + QT) .

motivation: entropy method operates mostly with quadratic functionals (e.g. e2(f |f∞); Iψ =

• ψ′′( f

f∞ )∇T f f∞ · D · ∇ f f∞ f∞ dx )

Lemma 3

Let Q be positively stable, i.e. μ := min{ℜ λQ} > 0 . ⇒ λmin(Qs) ≤ μ (“typically” strict inequality)

Lemma 4 ( = Lemma 2 for choice of P)

∃ P > 0 such that the similar matrix ˜ Q := √ PQ √ P

−1 satisfies:

λmin(˜ Qs) = μ ( λmin(˜ Qs) = μ − ε in the defective case )

• So: ∃ a similarity transformation such that Q, ˜

Q, ˜ Qs have the same spectral gap.

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 30 / 33

Application to ft = div(D∇f + Cxf ) = Lf

• Decay estimate of the drift characteristics ˙

x = −Cx: Let x2

P := x, Px .

d dt x2

P

= −2xT PCx = −( √ Px)T √ PC √ P

−1 +

√ P

−1CT √

P

• :=2˜

Cs≥2µI

• (

√ Px) ≤ −2μx2

P

• carries over to all invariant eigenspaces of L

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 31 / 33

Conclusion

new entropy method for degenerate Fokker-Planck equations (with linear drift) key tool: modified “entropy dissipation” functional → diff. inequal.

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 32 / 33

Conclusion

new entropy method for degenerate Fokker-Planck equations (with linear drift) key tool: modified “entropy dissipation” functional → diff. inequal. extension to kinetic Fokker-Planck equation (with non-quadratic potential) extension to non-degenerate, non-symmetric Fokker-Planck equations → sharp envelope for global decay work in progress: extension to BGK models [Achleitner-Arnold-Carlen]

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 32 / 33

Conclusion

new entropy method for degenerate Fokker-Planck equations (with linear drift) key tool: modified “entropy dissipation” functional → diff. inequal. extension to kinetic Fokker-Planck equation (with non-quadratic potential) extension to non-degenerate, non-symmetric Fokker-Planck equations → sharp envelope for global decay work in progress: extension to BGK models [Achleitner-Arnold-Carlen]

References

• A. Arnold, J. Erb: Sharp entropy decay for hypocoercive and

non-symmetric Fokker-Planck equations with linear drift, arXiv 2014

• F. Achleitner, A. Arnold, D. St¨

urzer: Large-time behavior in non-symmetric Fokker-Planck equations, Rivista di Matematica della

• Univ. di Parma, 2015.

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 32 / 33

References: [Villani] 2009: exponential decay in weighted H1, but no sharp rates [Dolbeault-Mouhot-Schmeiser] Trans. AMS 2014: kinetic models, exponential decay in modified L2–norm; m = 1 [Gadat-Miclo] KRM 2013: sharp rates for 2 Fokker-Planck toy models [Baudoin] 2014: Γ2–formalism, includes auxiliary gradient functional

Anton ARNOLD (TU Vienna) hypocoercive Fokker-Planck/entropy meth. 33 / 33